[Fornax96]

I'm not entirely sure on the specifics of storage proofs, but as far as I know it's something along these lines:

When uploading data the renter (that's what we call the node which pays for storage) computes a merkle tree of the data which the host should be storing. When a contract is nearing its end the host will enter a proof window of 144 blocks (1 full day) in which it will need to prove that it is storing the renter's data. The proof is probably based on the block hash of the block where the window started. The host stores the proof in the blockchain and the renter will be able to see the transaction. If the proof matches the merkle tree (which the renter has stored) the contract will end and the host will receive the payment and their collateral back. If the proof is invalid or was not submitted at all the renter can cancel the contract which destroys the funds in it. The host won't get paid and loses its collateral, but the renter also won't get their money back (to discourage the renter from playing foul)

There is some more info on this on the wiki: https://siawiki.tech/about/trustlessness and the website: https://sia.tech/technology. And here is some incomplete technical documentation: https://gitlab.com/NebulousLabs/Sia/-/blob/master/doc/Resour...

If you want to go more in-depth you can go on our Discord where lots of developers hang out, eager to help others to get started with the network :) https://discordapp.com/invite/sia

EDIT: The whitepaper is of course the best source of knowledge. It's quite old at this point but the core principles still apply https://sia.tech/sia.pdf